R E S E A R C H
Open Access
Askey-Wilson integral and its generalizations
Paweł J Szabłowski
**Correspondence:
[email protected] Department of Mathematics and Information Sciences, Warsaw University of Technology, ul. Koszykowa 75, Warsaw 00-662, Poland
Abstract
We expand the Askey-Wilson (AW) density in a series of products of continuous q-Hermite polynomials times the density that makes these polynomials orthogonal. As a by-product we obtain the value of the AW integral as well as the values of integrals ofq-Hermite polynomial times the AW density (q-Hermite moments of AW density). Our approach uses nice, old formulae of Carlitz and is general enough to venture a generalization. We prove that it is possible and pave the way how to do it. As a result, we obtain a system of recurrences that if solved successfully gives a sequence of generalized AW densities with more and more parameters.
MSC: 33D45; 05A30; 05E05
Keywords: Askey-Wilson integral; Askey-Wilson polynomials;q-Hermite polynomials; expansion of ratio of densities; symmetric functions
1 Introduction and preliminaries
1.1 Introduction
We consider a sequence of nonnegative, integrable functionsgn: [–, ]→R+defined by the formula
gn
x|a(n),q=fh(x|q) n
j=
ϕh(x|aj,q),
wherea(n)= (a
, . . . ,an), functionsfh andϕh defined by (.) and (.) denote respec-tively the density of measure that makes the so-called continuousq-Hermite polynomi-als orthogonal and the generating function of these polynomipolynomi-als calculated at pointsaj, j= , . . . ,n. Naturally functionsgnare symmetric with respect to vectorsa(n).
Our elementary but crucial for this paper observation is that examples of such functions are proportional to the densities of measures that make orthogonal respectively the so-called continuousq-Hermite (q-Hermite,n= [, Eq. (..)]), bigq-Hermite (bqH, n= [, Eq. (..)]), Al-Salam-Chihara (ASC,n= [, Eq. (..)]), continuous dual Hahn (CH,n= [, Eq. (..)]), Askey-Wilson (AW,n= [, Eq. (..)]) polynomials. This observation makes functionsgnimportant and, what is more exciting, allows possible generalization of both AW integral and AW polynomials,i.e., go beyondn= .
Similar observations were made in fact in [] when commenting on formula (..). Hence one can say that we are developing a certain idea of [].
Let us notice that this is a second attempt to generalize AW polynomials. The other one was made in [] by generalizing certain properties of generating functions ofq-Hermite, bqH, ASC, CH and AW polynomials.
On the other hand, by the observation that these functions are symmetric in variables
a(n)we enter the fascinating world of symmetric functions.
The paper is organized as follows. Next Section . presents notation that will be used and basic families of orthogonal polynomials that will appear in the sequel. We also present here important properties of these polynomials.
Section is devoted to expanding functionsgnin the series of the form
gn
x|a(n),q=An
a(n),qfh(x|q)
j≥
Tj(n)(a(n),q) (q)j
hj(x|q),
where{hn}denoteq-Hermite polynomials,{Tj(n)}are the sequences of certain symmetric functions and finally{An}are the values of the integrals
– gn
x|a(n),qdx,
and symbol (q)jis explained at the beginning of the next subsection.
We do this effectively forn= , . . . , , obtaining known results in a new way. In Section we show that sequences defined above do exist, and we present the way how to obtain them recursively. We are unable, however, to present nice compact forms of these sequences resembling those obtained forn≤, thus posing several open questions (see Section .) and leaving the field to younger and more talented researchers.
The partially legible, although not very compact, form was obtained for– g(x|a(),q)dx (see (.)).
Forq= , the case important for the rapidly developing so-called free probability, we give a simple, compact form for– g(x|a(), )dx(see Theorem (ii)) paving the way to conjecture the compact form of (.).
Tedious proofs are shifted to Section .
1.2 Preliminaries
qis a parameter. We will assume that – <q≤ unless otherwise stated. Let us define []q= , [n]q= +q+· · ·+qn–, [n]q! =
n
j=[j]q, with []q! = and
n k
q =
[n]q!
[n–k]q![k]q!, n≥k≥, , otherwise.
We will use the so-calledq-Pochhammer symbol forn≥,
(a;q)n= n–
j=
–aqj,
(a,a, . . . ,ak;q)n= n
j= (aj;q)n,
with (a;q)= .
It is easy to notice that (q)n= ( –q)n[n]q! and that
n k
q =
(q)
n
(q)n–k(q)k, n≥k≥,
, otherwise. (.)
The caseq= will be considered only when it might make sense and will be understood as the limitq→–.
Remark Notice that [n]=n, [n]! =n!,kn=nk, (a; )n= ( –a)nand [n]=
ifn≥, ifn= ,
[n]! = , n
k
= , for ≤k≤n, (a; )n=
ifn= , –a ifn≥.
We will need the following sets of polynomials.
The Rogers-Szegö polynomials that are defined by the equality
wn(x|q) = n
k= n k
q
xk (.)
forn≥ andw–(x|q) = . They will play an auxiliary role in the sequel. In particular one shows (see,e.g., []) that the polynomials defined by
hn(x|q) =einθwn
e–iθ|q, (.)
wherex=cosθ, satisfy the following -term recurrence:
hn+(x|q) = xhn(x|q) –
–qnhn–(x|q), (.)
withh–(x|q) = ,h(x|q) = .
These polynomials are the so-called continuousq-Hermite polynomials. A lot is known about their properties. For good reference, see [, ] or []. In particular we know that for |q|< ,
sup
|x|≤
hn(x|q)≤wn(|q).
Remark Notice thathn(x|) equals thenth Chebyshev polynomial of the second kind. More about these polynomials, one can find in,e.g., []. To analyze the caseq= , let us consider rescaled polynomialshn,i.e.,Hn(x|q) =hn(x√ –q/|q)/( –q)n/. Then equation (.) takes a form
Hn+(x|q) =xHn(x|q) – [n]qHn–(x|q),
In the sequel the following identities discovered by Carlitz (see Exercise .(b), (c) of []), true for|q|,|t|< ,
∞
k=
wk(|q)tk (q)k
= (t)
∞, ∞
k= w
k(|q)tk (q)k
=(t )
∞
(t)
∞ , (.)
will enable us to show absolute and uniform convergence of practically all series consid-ered in the sequel.
We have also the so-called linearization formula [, Eq. (..)], which can be dated back in fact to Rogers and Carlitz (see [, Eq. (..)] withβ= or [] for Rogers-Szegö polynomials), as follows:
hn(x|q)hm(x|q) =
min(n,m)
j= m
j
q n
j
q
(q)jhn+m–k(x|q), (.)
that will be our basic tool.
We will use the following two formulae of Carlitz presented in [] that concern proper-ties of Rogers-Szegö polynomials. Let us define two sets of functions
ζn(x|a,q) =
m≥ am (q)m
wn+m(x|q),
λn,m(x,y|a,q) =
k≥ ak (q)k
wn+k(x|q)wm+k(y|q),
defined for|x|,|y| ≤,|a|< andn,mbeing nonnegative integers. Carlitz proved ([, Eq. (.)], after correcting an obvious misprint) that
ζn(x|a,q) =ζ(x|a,q)μn(x|a,q), (.)
ζ(x|a,q) =
(a,ax)∞, (.)
where functionsμnare polynomials that are defined by
μn(x|a,q) = n
j= n
j
q
(a)jxj, (.)
and that ([, Eq. (.)], casem= also given in [, Exercise .(d)])
λm,n(x,y|a,q)
λ,(x,y|a,q) = m
j= n
k= n k
q m
j
q
(ax)j(ay)k(xya)k+j (xya)
k+j
xm–jyn–k, (.)
with
λ,(x,y|a,q) = (xya )
∞
(a,ax,ay,axy)∞. (.)
Proposition
xnμn
x–|a,q= n
j= n
j
q
(–a)jq(j)wn–j(x|q), (.)
wn(x|q) = n
k= n k
q
akxn–kμn–k
x–|a,q. (.)
To perform our calculations, we will also need the following two functions.
The generating function of theq-Hermite polynomials that is given by the formula below (see [, Eq. (..)]):
ϕh(x|t,q) df
=
j≥ tj (q)j
hj(x|q) = ∞
k=v(x|tqk)
, (.)
wherev(x|t) = – tx+t. Notice thatv(x|t)≥ for|x| ≤ and that from (.) it follows that series in (.) converges for|t|< . Notice also that from (.) it follows that
sup
|x|≤
ϕh(x|t,q) = /
|t|∞. (.)
The density of the measure with respect to which polynomialshnare orthogonal is given in,e.g., [, Eq. (..)]. Following it we have
–
hn(x|q)hm(x|q)fh(x|q)dx= (q)nδnm,
whereδmndenotes Kronecker’s delta, and
fh(x|q) =
(q)∞√ –x π
∞
k=
lx|qi, (.)
wherel(x|a) = ( +a)– ax. Notice that
sup
|x|≤
fh(x,q)≤(q)∞(–q)∞/π, (.)
following (.) sincel(x|q)≤( +q)for|x| ≤.
Remark We have
fh(x|) = √
–x/π, ϕ
h(x|a, ) = /
– ax+a
for|x|,|a|< .
After proper rescaling and normalization similar to the one performed in Remark , the caseq= leads to
forx,a∈R, as respectively the density of orthogonalizing measure and the generating function. For details, see [] or [].
2 Main results
Since in our approach symmetric polynomials will appear, let us introduce the following set of symmetric polynomials ofkvariables:
S(nk)(a, . . . ,ak|q) =
j,...,jk–≥
j+···+jk–≤n
j, . . . ,n– k–
m= jm
q aj
· · ·a jk–
k–a
n–j–···–jk–
k , (.)
where we denoted by [j,j, . . . ,n–km–=jm]qthe so-calledq-multinomial coefficient de-fined by [n, . . . ,nm]q= (q)n+···+nm/
m k=(q)nk.
Remark Notice thatS(nk)(a, . . . ,ak|) = ( k
j=aj)n.
Proof Obvious since (q)n k–
m=(q)jm(q)n–j–···–jk–|q==
n!
(n–km–=jm)!km–=jm!.
Proposition Let q∈(–, ),then:
(i)
n≥ tn (q)n
Sn(k)(a, . . . ,ak|q) = k j=(ait)∞
. (.)
(ii) For|t|< and∀j= , . . . ,k,
S(nk)(a, . . . ,ak|q) = n
m= n m
q
Sm(j)(a, . . . ,aj)S(nk––mj)(aj+, . . . ,ak|q). (.)
Ifq= ,then
n≥ tn n!S
(k)
n (a, . . . ,ak|) =exp
t k
j= aj
.
(iii)
Sn(k)(a, . . . ,ak|q)≤
max ≤j≤k|aj|
n
S(nk)(, . . . , |q).
Proof (i) Notice that
n≥ tn (q)n
Sn(k)(a, . . . ,ak|q) = n≥
j,...,jk–≥
j+···+jk–≤n
(ta)j· · ·(ta
k–)jk–(tak)n–j–···–jk– k–
m=(q)jm(q)n–j–···–jk
.
Secondly recall that (a)∞ =
j≥ a
j
(q)j. Now the assertion is easy. (ii) follows either from
Recall (i.e., [] or []) that there exist sets of orthogonal polynomials forming a part of the so-called AW scheme that are orthogonal with respect to measures with densities mentioned below. Although our main interest is in providing a simple proof of the so-called AW integral, we will list related densities for better exposition and for indicating the ways of possible generalization of AW integrals and polynomials.
So let us mention first the so-called bigq-Hermite polynomials{hn(x|a,q)}n≥–whose orthogonalizing measure has density for|a|< . The densityfbhof the orthogonalizing measure has a form (see [, Eq. (..)]) which can be written with the help of functions fhandϕh, namely
fbh(x|a,q) =fh(x|q)ϕh(x|a,q), (.)
–
hn(x|a,q)hm(x|a,q)fbh(x|a,q)dx= (q)nδmn. (.)
The form of polynomialshn(x|a,q) and their relation toq-Hermite polynomials is not important for our considerations. It can be found,e.g., in [, Eq. (..)] or in [, Eqs. (.), (.)]. So, for the sake of completeness, let us remark that from (.) it follows immediately that for|x| ≤,|a|< ,
fbh(x|a,q) =fh(x|q)
n≥ an (q)n
hn(x|q).
Here and below, where we will present similar expansions, convergence is almost uniform since all these expansions are in fact the Fourier series and that the Rademacher-Menshov theorem can be applied following (.).
Let us notice immediately that following (.) we have
–
hn(x|q)fbh(x|a,q)dx=an.
Secondly let us mention the so-called Al-Salam-Chihara polynomials{Qn(x|a,b,q)}n≥– that are orthogonal with respect to the measure that for|a|,|b|< has the density of the form (compare [, Eq. (..)])
fQ(x|a,b,q) = (ab)∞fh(x|q)ϕh(x|a,q)ϕh(x|b,q). (.)
We have the following lemma that illustrates our method and we will give a very simple proof of the well-known Poisson-Mehler formula as a corollary.
Lemma For|x| ≤,|a|,|b|< ,we have
fQ(x|a,b,q) =fh(x|q)
∞
j=
S()j (a,b) (q)j
hj(x|q). (.)
Proof Following (.) and (.) we have
fQ(x|a,b,q) = (ab)∞fh(x|q)
j,k≥ ajbk (q)j(q)k
Now we use (.) and (.) and change the order of summation to get
fQ(x|a,b,q) = (ab)∞fh(x|q)
m≥ (ab)m
(q)m
j,k≥m
aj–mbk–m (q)j–m(q)k–m
hj–k+m–k(x|q)
= (ab)∞fh(x|q)
m≥ (ab)m
(q)m
n,i≥ anbi (q)i(q)n
hn+i(x|q)
=fh(x|q)
s≥ hs(x|q)
(q)s s
n= s j
q anbs–n.
As an immediate corollary of our result, we have
–
hn(x|q)fQ(x|a,b,q)dx=Sn()(a,b|q). (.)
Remark Leta=ρeiη,b=ρe–iηand denotey=cosη. Then (i) S()n (a,b|q) =ρnhn(y|q),
(ii) v(x|a)v(x|b) = ( –ρ)– xyρ( +ρ) + ρ(x+y).
Proof (i) is an immediate consequence of (.). (ii) We havev(x|a)v(x|b) = ( – ρxeiη+ ρeiη)( – ρxe–iη+ρe–iη).
As a slightly more complicated corollary implied by Lemma , we have the following famous Poisson-Mehler (PM) expansion formula.
Corollary For|x|,|y|< ,|ρ|< ,we have
(ρ)∞
∞
k=( –ρqk)– xyρqk( +ρqk) + ρqk(x+y)
=
j≥
ρj
(q)j
hj(x|q)hj(y|q). (.)
Proof We takea=ρeiη,b=ρe–iηand denotey=cosη. Now we use (.) and Remark (ii)
to get the left-hand side multiplied byfh. Then we apply Lemma and Remark (i) to get the right-hand side of our PM formula also multiplied byfh. Finally we cancel outfhwhich
is positive on (–, ).
Remark The calculations we have performed while proving Lemma are very much
like those performed in [] while proving Theorem .. concerning the Poisson kernel (or Poisson-Mehler) formula. There exist many proofs of the PM formula (see,e.g., [] or a recently obtained very short one in []). In fact formula (.) can be dated back to Carlitz who in [] formulated it for Rogers-Szegö polynomials. The one presented above, which seems to be one of the shortest, was obtained as a by-product and, as it has already been mentioned, is almost the same as the one presented in [].
Notice that considering (.) witha=ρeiη,b=ρe–iηandy=cosηleads in view of
Re-mark (i) to
–
a nice symmetric formula that appeared in [] in a probabilistic context. Its probabilistic interpretation was exploited further in [].
Third in our sequence of families of polynomials that constitute AW scheme is the so-called continuous dual Hahn (CH) polynomials. Again their relationship to other sets of polynomials is not important. From [, Eq. (..)] it follows that the density of measure that makes them orthogonal is given by the following formula:
fCH(x|a,b,c,q) = (ab,ac,bc)∞fh(x|q)ϕh(x|a,q)ϕh(x|b,q)ϕh(x|c,q).
We have the following lemma.
Lemma
fCH(x|a,b,c,q) =fh(x|q)
n≥
σn()(a,b,c|q) (q)n
hn(x|q),
where
σn()(a,b,c|q) = n
j= n
j
q
q(j)(–abc)jS()
n–j(a,b,c|q). (.)
Proof Proof is shifted to Section .
Remark Notice that for|t|< ,
n≥ tn (q)n
σn()(a,c,b|q) = (abct)∞ (at,bt,ct)∞.
Proof Using (.) we have
n≥ tn (q)n
σn()(a,c,b|q) = n≥
tn (q)n
n
j= n
j
q
q(j)(–abc)jS()
n–j(a,b,c|q)
=
∞
j=
(–abct)j (q)j
q(j)
n≥j tn–j (q)n–j
S()n–j(a,b,c|q).
Now it remains to change the index of summation in the second sum, use (.) and use the fact that∞j=(–abct(q) )j
j q (j
)= (abct)∞.
Corollary For|a|,|b|,|c|< ,
–
hn(x|q)fCH(x|a,b,c,q)dx=σn()(a,b,c|q).
Proof Elementary.
of AW scheme are not important for our considerations. Recently a relatively rich study of these relationships was done in [], hence it may serve as the reference. We need only the form of AW density. It is given,e.g., in [, Eq. (..)] and, after necessary adaptation to our notation, is presented below,
fAW(x|a,b,c,d,q) =
(ab,ac,ad,bc,bd,cd)∞
(abcd)∞
×fh(x|a)ϕh(x|a,q)ϕh(x|b,q)ϕh(x|c,q)ϕh(x|d,q)
for|x| ≤,|a|,|b|,|c|,|d|< . Our main result concerns this density and is the following.
Theorem For|x| ≤,|a|,|b|,|c|,|d|< ,
fAW(x|a,b,c,d,q) =fh(x|q)
n≥
σn()(a,b,c,d|q) (q)n
hn(x|q), (.)
where
σn()(a,b,c,d|q) = n
j= n
j
q (bd)j (abcd)j
S()n–j(b,d|q)
× j
k= j k
q
(cb)kak(ad)j–kcj–k, (.)
are symmetric functions of a,b,c,d.
Proof Proof is shifted to Section .
As immediate corollaries we have the following fact.
Corollary Formax(|a|,|b|,|c|,|d|) < ,
–
hn(x)fAW(x|a,b,c,d,q)dx=σn()(a,b,c,d|q). (.)
Proof Follows directly from (.).
Remark Notice that from (.) in fact the value of AW integral follows since we see that– fAW(x|a,b,c,d|q)dx= , which means that the integral
π
– √
–x
n≥
l(x|qn)
v(x|aqn)v(x|bqn)v(x|cqn)v(x|dqn)dx
= (abcd)∞
(q,ab,ac,ad,bc,bd,cd)∞. (.)
Remark Notice also that (.) allows calculation of all moments of AW density. This is so since one knows the form of polynomialshn. Moments of AW density were calcu-lated by Corteel and Williams in in [] using combinatorial means. For complex a, b, c, d but forming conjugate pairs, this formula was also obtained independently about the same time. Namely it was done in [] where also an elegant expansion of
σn()(ρeiη,ρe–iη,ρeiθ,ρe–iθ|q) in terms ofhn(y|q) andhn(z|q), wherecosη=yandcosθ= z, was presented.
3 Generalization and open questions
3.1 Generalization
The results presented above allow the following generalization. The cases|q|< andq= will be treated separately. First let us consider|q|< .
Let us denotea(k)= (a, . . . ,a
k),k= , , . . . . We will assume that|x| ≤ and that all pa-rametersaihave absolute values less than . Let us denote
gn
x|a(n),q=fh(x|q) n
j=
ϕh(x|ai,q),
where functionsfhandϕhwere defined by (.) and (.) respectively. We remark following (.) and (.) that
gn
x|a(n),q≤(q)∞(–q)
∞ π
n
j= (|aj|)
∞
(.)
for|x| ≤, and|ai|< forj= , . . . ,n. We have the following general result.
Lemma For every n≥,there exist An(a(n),q)a symmetric function ofa(n)and a sequence
of symmetric ina(n)functions{Tj(n)(a(n),q)}j≥such that for|ak|< ,k= , . . . ,n,
gn
x|a(n),q=An
a(n),qfh(x|q)
j≥
Tj(n)(a(n),q) (q)j
hj(x|q). (.)
Moreover,
j≥
Tj(n)a(n),q<∞. (.)
Proof Let G = L(–, ,F,fh) be the space of functions h: –, → R such that
–h (x)f
h(x|q)dx. Notice that this space is spanned by the polynomials {hj(x|q)}j≥. Visibly, under our assumptions and by (.), nj=ϕh(x|ai,q)∈ G. Now notice that {Tj(n)(a(n),q)}j≥are coefficients of the Fourier expansion of the function
n
j=ϕh(x|ai,q) inGwith respect to{hj(x|q)}j≥. Since
– fh(x|q)
j≥
Tj(n)(a(n),q) (q)j
An(a(n),q) is the value of
–gn(x|a(n),q)dx. Inequality (.) follow properties of the Fourier expansion, more precisely Perseval’s identity. The fact thatAnand{Tj(n)}j≥are symmetric follows the observations thatnj=ϕh(x|ai,q) is symmetric.
Using formula (.) we can writegnin the following way wherehjareq-Hermite poly-nomials defined by (.). FunctionsAn(a(n),q) and{Tj(n)(a(n),q)}j≥have the following in-terpretation:
[–,]
hj(x|q)gn
x|a(n),qdx=An
a(n),qTj(n)a(n),q
forn,j≥.
We have the following easy proposition giving recursions that are satisfied by functions AnandTj(n).
Proposition Let us define a new sequence of functions{Hs(a(n),q)}n,s≥of n variables as follows:
m≥ am
n (q)m
Ts(+nm–)a(n–),q=Hs(n)a(n),q m≥
am n (q)m
Tm(n–)a(n–),q.
Then
(i)
An
a(n),q=An–
a(n–),q m≥
am n (q)m
Tm(n–)a(n–),q,
(ii)
Tj(n)a(n),q= j
s= j s
q
Hs(n–)a(n–),q(an)j–s.
Proof Proof is shifted to Section .
Remark The integral– gn(x|a(n),q)dxwas calculated in [] (see also Theorem .. in []) by combinatorial methods. The obtained formula is, however, very complicated. Be-sides, the above mentioned Theorem .. of [] does not provide expansion (.) which is automatically obtained in our approach.
Remark Notice also that following Proposition (i) we get for|aj|< ,j= , . . . , ,
– g
x|a(),qdx= (
j aj)∞ (q)∞≤k<m≤(akam)∞
j≥ aj (q)j
σj()(a,a,a,a|q). (.)
Forq= the calculations presented in (.) can be carried out completely and the con-cise form can be obtained. This is possible due to the following simplified form of (.).
(i)
σn()(a,a,a,a|)
=S()n (a,a|) +( –ad)( –aa) ( –aaaa) aS
()
n–(a,a,a|)
+( –aa)( –aa) ( –aaaa) aS
()
n–(a,a,a|)
+( –aa)( –aa)( –aa)aa ( –aaaa)
S()n–(a,a,a,a|),
(ii)
– g
x|a(), dx= –χ(a
()) +χ(a())χ(a()) –χ (a())
≤j<k≤( –ajak)
,
whereχ, . . . ,χdenote respectively first five elementary symmetric functions of the vectora().That is,χj(a(k)) =
≤n<n···<nj≤k
j m=anm.
Proof Proof is shifted to Section .
Forq= , the problem of finding sequencesAn(a(n)|) and{Tj(n)(a(n), )}j≥can be solved completely and trivially. Namely we have the following.
Proposition
An
a(n)|=exp ≤j<k≤n
ajak
,
Tj(n)a(n), = n
k= ak
j .
Proof Using Remark we get
gn
x|a(n), =exp
–x/ +x n
j= aj–
n
j= aj
√ π
=√ πexp
n
j= aj
–
n
j= aj
×exp
–x/ +x n
j= aj–
n
j= aj
=exp ≤j<k≤n
ajak
exp(–x/) √
π
j≥
(nk=ak)j
j! Hj(x).
3.2 Unsolved problems and open questions
.. Questions
• What are the compact forms of functions{Tj(n)(a(n),q)}j
• What are the compact forms of these functions forq= (free probability case)? • Following formula for– g(x|a(), )dxgiven in assertion (ii) of Theorem is it true
that
– g
x|a(),qdx=(χ(a
()) –χ(a())χ(a()) +χ (a()))∞
≤j<k≤(ajak)∞ ?
Notice that fora= it would reduce to AW integral.
• It would be valuable to get values{An(a(n),q)}forn= , and so on for complex
values of parametersa(n)but forming conjugate pairs. It would be also fascinating to
find polynomials that would be orthogonalized by densities obtained in this way. This problem follows the probabilistic interpretation of Askey-Wilson density rescaled with complex parameters. Such an interpretation of finite Markov chains of length at leastwas presented in [, ]. Let{X,X,X}denote this finite Markov
chain. Recall that then AW density can be interpreted as the conditional density of
X|X,X.
It would be exciting to find out if, for sayn= , a similar probabilistic interpretation could be established. That is, if we could define five-dimensional random vector
(X, . . . ,X)with normalized functiong(x|a(),q)as the conditional densityX|X,X,
X,X. Note that then the chain(X, . . . ,X)could not be Markov. Similar questions apply to the casen= , , . . ..
.. Unsolved related problems and direction of further research
. In [] we find Theorem .. which is due to Gasper and Rahman () and which can be stated in our notation. Formax≤j≤|aj|< ,|q|< , we have
–
g(x|a(),q)
ϕh(x|
j=aj,q) dx=
j=(
k=,k=jak)∞
≤j<k≤(ajak)∞ .
This result suggests considering the following functions:
Gn,m
x|a(n),b(m),q=fh(x|q) n
j=ϕh(x|ai,q) m
k=ϕh(x|bk,q) ,
wherea(n)andb(m)are certain vectors of dimensions respectivelynandm, find its integrals over[–, ]and expansions similar to (.).
. Recently a paper [] onq-Laplace transform, where many analogies to ordinary case were indicated, has appeared. What would aq-Laplace transform of the distributions that were considered above be?
4 Proofs
Proof of Lemma We have
k,n,m≥ an (q)n
bm (q)m
ck (q)k
hn(x|q)hm(x|q)hk(x|q)
= (ab)∞
j≥ hj(x|q)
(q)j
∞
m= cm (q)m
j
k= j k
q
Since obviouslyS()n (a,b|q) =anwn(b/a|q), we get
Now we apply formula (.) and get
Now we use (.) and Proposition (ii) and get
Proof of Theorem Applying (.) we get
×
Now we apply Carlitz formulae (.) and (.) to get
This fact is a result of the following calculations:
while the right-hand side to
=
Proof of Theorem We use (.) and utilizing Remark we get
σn()(a,b,c,d|) =S()n (b,d|) + ( –bd) formulae (.) and (.) remembering that ()n= leads to our integral formula.
Competing interests
Acknowledgements
The author is very grateful to an unknown referee for pointing out additional references and just evaluation of the paper.
Received: 25 August 2014 Accepted: 26 November 2014 Published:15 Dec 2014 References
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Cite this article as:Szabłowski:Askey-Wilson integral and its generalizations.Advances in Difference Equations
